Fundamentals of Plasma Physics

5.2 The Landau problem 153

are worked out in the assignments showing that the maximum growth rate is

maxωi≃

√

3

2

(

me

2 mi

) 1 / 3

ωpe (5.18)

which occurs when
k·u 0 ≃ωpe. (5.19)
Again this is a very fast growing instability, about one order of magnitude smaller than the
electron plasma frequency.
Streaming instabilities are a reason why certain simple proposed methods for attaining
thermonuclear fusion will not work. These methods involve shooting an energetic deu-
terium beam at an oppositely directed energetic tritium beam with theexpectation that
collisions between the two beams would produce fusion reactions. However, such a system
is extremely unstable with respect to the two-stream instability. This instability typically
has a growth rate much faster than the fusion reaction rate and so will destroy the beams
before significant fusion reactions can occur.

5.2 The Landau problem

A plasma wave behavior that is both of great philosophical interest and great practical
importance can now be investigated. Before doing so, three seemingly disconnected results
obtained thus far should be mentioned, namely:

1. When the exchange of energy between charged particles and a simple one-dimensional
   wave having dependence∼exp(ikx−iωt)was considered, the particles were catego-
   rized into two general classes, trapped and untrapped, and it was found that untrapped
   particles tended to be dragged toward the wave phase velocity. Thus, untrapped par-
   ticles moving slower than the wave gain kinetic energy, whereas those moving faster
   lose kinetic energy. This has the consequence that if there are more slow than fast
   particles, the particles gain net kinetic energy overall and this gain presumably comes
   at the expense of the wave. Conversely if there are more fast than slowparticles, net
   energyflows from the particles to the wave.


2. When electrostatic plasma waves in an unmagnetized, uniform, stationary plasma
   were considered it was found that wave behavior is characterized by a dispersion re-
   lation1+χe(ω,k)+χi(ω,k)=0,whereχσ(ω,k)is the susceptibility of each species
   σ.These susceptibilities had simple limiting forms whenω/k <<

√

κTσ 0 /mσ(isother-

mal limit) and whenω/k >>

√

κTσ 0 /mσ(adiabatic limit), but thefluid analysis

failed whenω/k∼

√

κTσ 0 /mσand the susceptibilities became undefined.

3. When the behavior of interacting beams of particles was considered, itwas found that
   under certain conditions a fast growing instability would develop.
   These three results will be tied together by the analysis of the Landau problem.

5.2.1 Attempt to solve the linearized Vlasov-Poisson system of equations using

Fourier analysis

The method for manipulatingfluid equations to find wave solutions was as follows: (i)